# Powering-invariance is not quotient-transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup)notsatisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property).

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## Contents

## Statement

It is possible to have groups such that is a powering-invariant normal subgroup of and is a powering-invariant subgroup of the quotient group , but is not powering-invariant in .

## Related facts

- Powering-invariant over quotient-powering-invariant implies powering-invariant
- Quotient-powering-invariance is quotient-transitive

## Proof

The proof idea is follows: use the construction in the reference for . Now take as a subgroup containing such that is a finite cyclic group of order . Now:

- is powering-invariant in by construction, since both and are rationally powered groups.
- is powering-invariant in since is not powered over
*any*prime. - is not powering-invariant in : For instance, an element of whose image in generates the latter group does not have a root in .